Yes, these words are connected. Chanukah means inauguration, as Chanukah celebrates the "Chanukat Hamizbe'ach," the re-inauguration of the altar by the Maccabees after its defilement.
Chinuch is an expression indicating the beginning of something. Thus, it means inauguration, but it also means education, which begins and initiates a person in the way that it is hoped that he will follow. As King Solomon wrote, "Chanoch l'na'ar al pi darko — teach a child according to his way...."
Chanukah when read as two words (chanu kaf-hey) means "they encamped on the 25th," indicating that that the Maccabees were victorious in battle and rested from their enemies on the 25th of Kislev.
